Trying to remember the title/author of a book that had a cat on the cover, with some planets or moons in the background. Borrowed it from my elementary school library in the 80's. The main character was a kid who grew up without a mother, and somehow ends up finding a gate that leads off the world (don't think it was set on Earth). The ending of the book implied that the kid's mother had disappeared through one of these gates, and that the kid was going to go through the gate and look for his mother (cliffhanger ending). Don't remember what the cat on the cover had to do with anything.
The Chronicles of Chrestomanci by Diana Wynne Jones?
Your description of the plot does not quite match my memory (nor the brief synopses that I have found), but there are several similarities that I think make this worth posting.
The cover matches amazingly, the cat is not (directly) a part of the story, a lot of the stories happens on alternate earths, etc.
There are several books in the series, with ones being published in 1977, 1980, and 1988 that would possibly be at an elementary school library.
Unfortunately the cover art I posted is from a US edition from 2007, but I imagine there were similar-ish ones in the '80s.
Obviously the stories in the different books are not the same, but the overarching "universe" is that there are many parallel worlds that are slightly different- different versions of Earth, and some of the characters learn how to visit them. Magic is totally a thing, and the "Chrestomanci" character (there are multiple) has 9 lives (hence the cat on the cover) and can learn to be very powerful in magic.
The Gate of the Cat by Andre Norton?
A stone ring in the Scottish Highlands becomes the gateway through which Kelsie McBlair enters a new world and becomes embroiled in a struggle beween Light and Dark for power over the Green Valley. Norton's many Witch World fans will welcome this return to a land of powerful witches, mail-clad warriors, and mysterious Old Ones
It's part of the Witch World series.